Two-dimensional photonic crystal, and waveguide and resonator using the same

ABSTRACT

A tetragonal lattice  104  is formed by cylindrical structural members  101,  and a photonic crystal  100  has a periodical structure formed by a periodical arrangement of such tetragonal lattice  104.  A distance between center points of the cylindrical structural members  101  is taken as a unit length a, which constitutes a lattice constant of the tetragonal lattice  104.  At an approximate center of the tetragonal lattice  104,  a cylindrical structural member  102  is provided, and a dielectric area  103  is provided around the cylindrical structural members  101  and the cylindrical structural member  102.  This structure allows to form a photonic band gap for a TE wave and a photonic band gap for a TM wave in a certain common frequency region, thereby forming a complete band gap.

TECHNICAL FIELD

The present invention relates to a photonic crystal adapted for use in adevice for controlling an electromagnetic wave such as light, and moreparticularly to a two-dimensional photonic crystal capable of forming acomplete photonic band gap to an electromagnetic wave of a specifiedwavelength region, and a waveguide and a resonator utilizing the same.

BACKGROUND ART

Recently, photonic crystals are becoming increasingly important asdevices for controlling electromagnetic waves such as light. A photoniccrystal is a periodical structured member showing a periodical change ina dielectric constant in crystal-constituting regions, with a periodicaldielectric change comparable to a wavelength of an electromagnetic wavesuch as light, and can realize novel electromagnetic characteristics byan artificial periodical structure. Such structure is featured, like aband gap formation in a semiconductor substance by a Bragg reflection ofelectrons by periodical potentials of atomic nuclei, by a formation ofband gap to an electromagnetic wave such as light, as suchelectromagnetic wave is subjected to a Bragg reflection by periodicaldistribution of refractive index. In a photonic crystal, such band gapis called a photonic band gap. Such photonic band gap, in which anelectromagnetic wave such as light cannot exist, allows to arbitrarilycontrol the electromagnetic wave such as light.

A band gap which inhibits a propagation of an electromagnetic wave suchas light in all the directions is called a complete band gap. In casesuch complete band gap is made possible, an ultra small device can beprepared in a photonic crystal, by forming a point defect or a lineardefect therein. For example, in case of artificially perturbing a partof a periodicity in the photonic crystal, a defect level is formed inthe photonic band gap and an electromagnetic wave such as light isallowed to exist only in such defect level, and such phenomenon can beutilized for example in a resonator. Also in case of forming a lineardefect, an electromagnetic wave such as light can propagate along anarray of defects but cannot propagate in other areas than the defects,so that an ultra small waveguide can be formed.

Therefore, in order to exploit the characteristics of the photonic bandgap, it is necessary to prepare a photonic crystal having a completeband gap.

As a photonic crystal structure having a wide complete band gap, thereis known a photonic crystal having a three-dimensional periodicalstructure (hereinafter represented as three-dimensional photoniccrystal) such as a Yablonovite structure (for example PatentLiterature 1) or a Woodpile structure (for example Non-Patent Literature1). Such crystals have a wide complete band gap, but are very difficultto produce because of structures thereof Also in case one of the pluraldielectric substances constituting the photonic crystal is air, thethree-dimensional periodical structure cannot be maintained when thedielectric substances are arranged three-dimensionally and in anon-contact manner as in a certain diamond or opal structure.

On the other hand, a photonic crystal having a two-dimensionalperiodical structure (hereinafter represented as two-dimensionalphotonic crystal) is easier to prepare in comparison with thethree-dimensional photonic crystal. For example, as a two-dimensionalphotonic crystal having a complete band gap, there is known atwo-dimensional photonic crystal constituted of a triangular latticearrangement formed by a circular hole (for example Patent Literature 2).Also as a relatively easily produceable structure, a two-dimensionalphotonic crystal having a tetragonal lattice arrangement formed bycircular holes or cylinders is known.

Also the photonic crystal is formed from two or more dielectricsubstances. Ordinarily there are employed two substances, one of whichis often air because of an ease in manufacture and a low loss. Forexample, in the aforementioned two-dimensional photonic crystalconstituted of a trigonal lattice structure or a tetragonal latticestructure, such trigonal lattice or tetragonal lattice is formed by air.

Patent Literature 1: U.S. Pat. No. 5,172,267

Patent Literature 2: JPA No. 2001-272555 (paragraph [0023], FIG. 1)

Non-Patent Literature 1: E. Knobloch, A. Deane, J. Toomre and D. R.Moore, Contemp. Math., 56, 203 (1986).

However, in the two-dimensional photonic crystal constituted of atrigonal lattice arrangement as described in Patent Literature 2, awidest complete band gap is obtained for r/a of 0.48 (wherein r is aradius of a circular hole and a is a lattice constant of the photoniccrystal). Therefore, a thickness between the circular holes becomes assmall as 0.04 a, and such photonic crystal is very difficult to prepare.

Also a two-dimensional photonic crystal constituted of a tetragonallattice structure, in case the tetragonal lattice is formed by circularholes, shows a band gap to a TE wave (transverse electric wave) of theincident electromagnetic wave but does not show a band gap to a TM wave(transverse magnetic wave). On the other hand, in case the tetragonallattice is formed by cylinders, it shows a band gap to the TM wave butdoes not show a band gap to the TE wave. Therefore, a complete band gapcannot be obtained in the two-dimensional photonic crystals constitutedby such tetragonal lattices.

Thus, there is required a two-dimensional photonic crystal that can beprepared easily and that shows band gaps to both the TE wave and the TMwave in all the incident angles in order to obtain a complete band gap.

On the other hand, as the photonic crystal is generally prepared by asemiconductor manufacturing technology and a photoforming technology,materials to be employed are limited to semiconductor materials andphotosettable resins. These materials have relatively small relativedielectric constants, so that a wide band gap is difficult to obtain. Amethod of mixing a ceramic powder in a photosettable resin is alsoknown, but a high relative dielectric constant cannot be obtained as therelative dielectric constant is governed by a logarithmic mixing ruleand is principally influenced by the relative dielectric constant of theresin, so that a wide band gap is difficult to obtain.

DISCLOSURE OF INVENTION

The present invention is to solve the aforementioned difficulties, andan object thereof is to provide a two-dimensional photonic crystal whichis easy to prepare and has a complete band gap to the TE wave and the TMwave in all the incident angles, and a waveguide and a resonatorutilizing the same.

Another object of the invention is to provide a two-dimensional photoniccrystal having a wide complete band gap utilizing a single ceramicmaterial of a high relative dielectric constant, and a waveguide and aresonator utilizing the same.

Still another object of the invention is to increase the strength of thetwo-dimensional photonic crystal itself by utilizing a dielectricmaterial other than air, thereby providing a two-dimensional photoniccrystal that can be prepared easily. Also the invention, by utilizing adielectric material other than air, enables to form a wider completeband gap and to achieve a relatively small size of a device utilizingthe two-dimensional photonic crystal. Also the invention is to provide atwo-dimensional photonic crystal capable of easily opening a band gapand reducing the loss by employing a ceramic material of a high relativedielectric constant, and a waveguide and a resonator utilizing the same.

A two-dimensional photonic crystal of the invention is a two-dimensionalphotonic crystal formed by a periodical two-dimensional arrangement ofplural unit lattices, characterized in including a prism-shaped firstdielectric area arranged in each lattice point of the unit lattice, aprism-shaped second dielectric area arranged at an approximate center ofthe unit lattice, and a third dielectric area adjacent to and around thefirst and second dielectric areas.

Also the two-dimensional photonic crystal of the invention ischaracterized in that the third dielectric area has a relativedielectric constant different from relative dielectric constants of thefirst and second dielectric areas.

In the present invention, the unit lattice is preferably a tetragonallattice.

Also the first dielectric area and the second dielectric area preferablyhave a substantially cylindrical shape and satisfy a relationship:0.4 a≦r 1+r 2≦0.6 awherein r1 indicates a radius of the cylindrical first dielectric area,r2 indicates a radius of the cylindrical second dielectric area, and aindicates a unit length of a lattice axis of the tetragonal lattice.

Also a relative dielectric constant ε1 of the first dielectric area maybe equal to or smaller than a relative dielectric constant ε2 of thesecond dielectric area.

Also a relative dielectric constant ε3 of the third dielectric areapreferably satisfies at least a relation ε3>ε1.

Further, a relative dielectric constant ε1 of the first dielectric area,a relative dielectric constant ε2 of the second dielectric area, and arelative dielectric constant ε3 of the third dielectric area preferablysatisfy relations:ε3>ε1 and ε2/ε1>20.

The first and second dielectric areas may be formed by air and the thirddielectric area may be formed by a dielectric material containing aceramic material.

Also the first, second and third dielectric areas may be formed by adielectric material containing a ceramic material.

Also a unit length a of the lattice axis of the tetragonal lattice ispreferably different depending on a frequency of a light or anelectromagnetic wave entering the two-dimensional photonic crystal.

Also a photonic crystal waveguide of the present invention ischaracterized in including the aforementioned two-dimensional photoniccrystal in which a linear defect is formed in a periodical latticearrangement of the two-dimensional photonic crystal.

Also a photonic crystal resonator of the present invention ischaracterized in including the aforementioned two-dimensional photoniccrystal in which a point-shaped defect is formed in a periodical latticearrangement of the two-dimensional photonic crystal.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a perspective view showing a photonic crystalline structureembodying the present invention.

FIG. 2 is a plan view showing an arrangement of dielectric areas in anembodiment of the invention.

FIG. 3 is a table showing a result of simulation on a photonic crystalin a first embodiment of the invention.

FIG. 4 is a table showing a result of simulation on a photonic crystalin a second embodiment of the invention.

FIG. 5 is a table showing a result of simulation on a photonic crystalin a third embodiment of the invention.

FIG. 6 is a view showing a manufacturing process of a photonic crystalin a first embodiment of the invention.

FIG. 7 is a view showing a manufacturing process of a photonic crystalin a second embodiment of the invention.

FIG. 8 is a view showing a manufacturing process of a photonic crystalin a third embodiment of the invention.

FIG. 9 is a schematic view showing a photonic crystal in anotherembodiment.

FIG. 10 is a schematic view showing a photonic crystal in still anotherembodiment.

BEST MODE FOR CARRYING OUT THE INVENTION

In the following, embodiments of the present invention will be explainedwith reference to FIGS. 1 to 8. FIG. 1 is a perspective view showing aphotonic crystalline structure embodying the present invention, and FIG.2 is a plan view for explaining an arrangement of dielectric areas in anembodiment of the invention. Also FIGS. 3 and 6 relate to a firstembodiment of the invention; FIGS. 4 and 7 relate to a second embodimentof the invention; and FIGS. 5 and 8 relates to a third embodiment of theinvention.

(Terms)

At first there will be explained terms to be used in the presentdescription.

A “two-dimensional photonic crystal” means a periodical structuredmember having a two-dimensional periodical structure of dielectricconstant in a direction substantially parallel to a specified plane.

Also a “unit lattice” means a structural member formed by a minimumperiodical unit constituting the two-dimensional photonic crystal, has atwo-dimensional shape formed by connecting structural members ordielectric areas positioned at outermost corners.

Also a “complete band gap” means a state capable of attenuating anelectromagnetic wave such as a light entering the two-dimensionalphotonic crystal in a direction substantially parallel to theaforementioned specified plane, to a predetermined amount or less, forall the incident angles, and more specifically means a band gap capableof attenuating the incident electromagnetic wave to an extent capable ofpreparing an ultra small device such as a resonator or a waveguide byforming a point defect or a linear defect.

(Configuration)

In the following, there will be explained, with reference to FIGS. 1 and2, a configuration of the photonic crystal embodying the presentinvention.

As shown in FIG. 1, a photonic crystal 100 embodying the presentinvention is constituted of plural first cylindrical structured members101, plural second cylindrical structured members 102, and a dielectricarea 103 provided around the cylindrical structured members 101 and thecylindrical structured members 102. The first cylindrical structuredmember 101 constitutes a first dielectric area, and the secondcylindrical structured member 102 constitutes a second dielectric area.Owing to these components, the photonic crystal 100 has atwo-dimensional periodical structure. In the following the structure ofthe present embodiment will be explained in more details.

As shown in FIG. 2, the first cylindrical structural members 101constitute a tetragonal lattice 104, and the photonic crystal of thepresent embodiment has a periodical structure in which such tetragonallattice 104 is periodically arranged. Therefore, in the photonic crystal100, the tetragonal lattice 104 constitutes a unit lattice. Also a sideof the tetragonal lattice 104 has a length “a”, which is taken as alattice constant. Also the second cylindrical structural member 102 isprovided at an approximate center of each tetragonal lattice 104, andthe dielectric area 103 is provided around the first cylindricalstructured member 101 and the second cylindrical structured member 102.FIG. 2 shows only a part of the photonic crystal of the presentinvention, but in practice the structure shown in FIG. 2 is periodicallyarranged.

The first cylindrical structured member 101 and the second cylindricalstructured member 102 are so constructed as to assume a substantiallycylindrical shape, with a radius r1 in the first cylindrical structuredmember 101 and a radius r2 in the second cylindrical structured member102. Also a relative dielectric constant is taken as ε1 in the firstcylindrical structural member 101, ε2 in the second cylindricalstructural member 102, and ε3 in the third dielectric area 103.

A complete band gap can be realized by varying the dielectric constantε1 and the radius r1 of the first cylindrical structural member 101, thedielectric constant ε2 and the radius r2 of the second cylindricalstructural member 102, and the dielectric constant ε3 of the thirddielectric area 103 under a suitable condition. Specific structures willbe explained in following examples.

Also the first dielectric area, the second dielectric area and the thirddielectric area can be constituted by a ceramic material, asemiconductor material, a resin or the like, or by air. As a ceramicmaterial, there can be employed, for example, a material based onBaO—TiO₂, BaO—Nd₂O₃—TiO₂, TiO₂, or Al₂O₃. Also as a semiconductormaterial, there can be employed, for example, GaAs, Si or SiO₂. Also asa resinous material, there can be emplolyed a polymer material of a highdielectric constant such as a polyvinylidene fluoride resin, a melanineresin, a urea resin, or a polyvinyl fluoride resin.

(Function)

The photonic crystal of the present embodiment having the aforementionedcrystal structure can exhibit following advantageous effects.

As a two-dimensional photonic crystal structure is formed by employing atetragonal lattice with first dielectric areas at the lattice points asa unit lattice, also providing a second dielectric area at anapproximate center thereof, and providing a third dielectric areaadjacent thereto, a photonic band gap to a TE wave and a photonic bandgap to a TM wave are formed in a certain common frequency region for allthe incident angles, thereby realizing a complete band gap.

Also the use of a dielectric material other than air for the firstdielectric area and the second dielectric area enables preparation of atwo-dimensional photonic crystal which is easy to prepare and which hasa wide complete band gap. A dielectric material other than air increasesa loss in comparison with the case of using air, but such loss can bealleviated by utilizing a ceramic material having a high relativedielectric constant, thus leading to a low loss.

EXAMPLES

In the following, the photonic crystal of the present embodiment will beexplained by specific examples.

First Example

At first there will be explained a first example. In a photonic crystal100 in the first example, a portion where the first cylindricalstructural member 101 exists as the first dielectric area and a portionwhere the second cylindrical structural member 102 exists as the seconddielectric area are vacant and are constituted by air. Therefore thefirst cylindrical structural member 101 and the second cylindricalstructural member 102 have relative dielectric constants ε1=ε2=1.0. Alsothe third dielectric area 103 is constituted of a material having arelative dielectric constant ε3 of 10.4.

In the first example, a width of the complete band gap was calculated bya simulation conducted by varying the radius r1 of the first cylindricalstructural member 101 and the radium r2 of the second cylindricalstructural member 102. More specifically, the simulation was conductedby changing the cylindrical radius r1 and the cylindrical radius r2respectively within a range of 0.1 a to 0.5 a for calculating the widthof the complete band gap. Results of calculation are summarized in FIG.3.

In the table shown in FIG. 3, the width of the complete band gap isrepresented in the unit of %. A detailed calculation method for thewidth of the complete band gap will be explained later, but a width (%)of the complete band gap shown in the table is represented, in case acomplete band gap is present continuously for a certain frequency range,by a value obtained by dividing such frequency range with a centralfrequency of such frequency range. In the table shown in FIG. 3, acolumn with “−” means that a complete band gap was not formed for thecylindrical radius r1 and the cylindrical radius r2 corresponding tosuch column. As a result of the simulation, in case of a cylindricalradius r1 of 0.1 a, the complete band gap showed a width of 6.39% for acylindrical radius r2 of 0.35 a, and a width of 2.70% for a cylindricalradius r2 of 0.40 a. Also in case of a cylindrical radius r1 of 0.2 a,the complete band gap showed a width of 2.48% for a cylindrical radiusr2 of 0.25 a, a width of 2.65% for a cylindrical radius r2 of 0.3 a, anda width of 2.60% for a cylindrical radius r2 of 0.35 a. Also in case ofa cylindrical radius r1 of 0.3 a, the complete band gap showed a widthof 4.07% for a cylindrical radius r2 of 0.2 a, a width of 2.23% for acylindrical radius r2 of 0.25 a, and a width of 1.96% for a cylindricalradius r2 of 0.3 a. Also in case of a cylindrical radius r1 of 0.4 a,the complete band gap showed a width of 2.70% for a cylindrical radiusr2 of 0.05 a, and a width of 1.81% for a cylindrical radius r2 of 0.1 a.

Therefore, a complete band gap could be obtained within a range wherethe radius r1 of the first cylindrical structural member 101 and theradius r2 of the second cylindrical structural member 102 satisfy arelation:0.4 a≦r 1+r 2≦0.6 a.

Also the material to be employed is not particularly restricted, and anymaterial capable of showing a relative dielectric constant ε3=10.4 inthe third dielectric area 103 can provide similar results. A ceramicmaterial is advantageous, but a semiconductor material or a resin may beemployed as long as a relative dielectric constant ε3=10.4 can beattained. Further, similar results will be presumably attainable evenwhen the relative dielectric constant ε3 is not 10.4. For example, incase of employing a ceramic material such as sapphire, a wide band gapwill be attainable because sapphire has a relative dielectric constant εof 9.4. Also the material is not limited to such ceramic material butcan also be a semiconductor material such as GaAs. GaAs, having arelative dielectric constant ε of 12 to 13, is considered to provide awide band gap as in the ceramic material.

Then, a complete band gap can be formed by changing the lattice constanta according to the frequency of the electromagnetic wave to be employed(wavelength of the electromagnetic wave). For example, in case acomplete band gap can be formed by adopting a lattice constant a=0.5 mmfor an electromagnetic wave of a frequency of 300 GHz as an example of amillimeter wave, a complete band gap can be realized for anelectromagnetic wave of a frequency of 3 GHz as an example of amicrowave by adopting a lattice constant a=50 mm. In this manner, acomplete band gap can be obtained for the electromagnetic wave from amillimeter wave region to a microwave region by suitably changing thelattice constant a of the photonic crystal of the invention according tothe electromagnetic wave to be employed. Also the lattice constant for agiven frequency can be made smaller by increasing the relativedielectric constant ε of the material to be utilized. Therefore, in casethe photonic crystal of the invention is applied to a device such as awaveguide or a resonator, such device can be made smaller in dimensionby employing a material of a higher relative dielectric constant.

Second Example

In the following, there will be explained a second example. In aphotonic crystal 100 in the second example, the first cylindricalstructural member 101 constituting a first dielectric area isconstituted of a material having a relative dielectric constant ε1=10,while the second cylindrical structural member 102 has a radius r2 sameas a radius r1 of the first cylindrical structural member 101. Also thethird dielectric area 103 is vacant and is constituted by air. Thereforethe third dielectric area 103 has a relative dielectric constant ε3=1.0.

In the second example, a width of the complete band gap was calculatedby a simulation conducted by varying the radius r1 of the firstcylindrical structural member 101 and the radium r2 of the secondcylindrical structural member 102. More specifically, the simulation wasconducted by changing the cylindrical radius r1 (=cylindrical radius r2)within a range of 0.2 a to 0.3 a and by changing the relative dielectricconstant ε2 within a range of 10 to 50 for calculating the width of thecomplete band gap. Results of calculation are summarized in FIG. 4. Inthe table shown in FIG. 4, a column with “−” means that a complete bandgap was not formed for the cylindrical radius r1 and the relativedielectric constant ε2 corresponding to such column.

As a result of the simulation, in case of a cylindrical radius r1 of 0.2a, the complete band gap showed a width of 2.65% for a relativedielectric constant ε2 of 20, also a width of 2.84% for a relativedielectric constant ε2 of 30, a width of 1.35% for a relative dielectricconstant ε2 of 40, and a width of 2.72% for a relative dielectricconstant ε2 of 50. Also in case of a cylindrical radius r1 of 0.25 a,the complete band gap showed a width of 1.03% for a relative dielectricconstant ε2 of 20, a width of 2.05% for a relative dielectric constantε2 of 30, a width of 1.72% for a relative dielectric constant ε2 of 40,and a width of 1.88% for a relative dielectric constant ε2 of 50. Alsoin case of a cylindrical radius r1 of 0.3 a, the complete band gapshowed a width of 1.96% for a relative dielectric constant ε3 of 40, anda width of 5.61% for a relative dielectric constant ε3 of 50.

Therefore, a complete band gap could be obtained within a range wherethe radius r1 of the first cylindrical structural member 101 and theradius r2 of the second cylindrical structural member 102 satisfy arelation:0.40 a≦r 1+r 2≦0.60 a,and where the relative dielectric constant ε1 of the first cylindricalstructural member 101 and the relative dielectric constant ε2 of thesecond cylindrical structural member 102 satisfy a relation:ε2/ε1≧2.

Also as in the first example, a complete band gap can be obtained withina frequency range from the millimeter wave to the microwave, by varyingthe lattice constant a of the photonic crystal according to thefrequency (wavelength) of the incident electromagnetic wave.

Third Example

In the following, there will be explained a third example. In a photoniccrystal 100 in the third example, a portion of the first cylindricalstructural member 101 constituting a first dielectric area is vacant andis constituted of air. Therefore, the portion of the first cylindricalstructural member 101 has a relative dielectric constant ε1=1.0. Also asecond cylindrical structural member 102 has a radius r2 same as aradius r1 of the first cylindrical structural member 101. Also the thirddielectric area 103 is constituted by a material of a relativedielectric constant ε3=10.4.

In the third example, a width of the complete band gap was calculated byvarying the radius r1 of the first cylindrical structural member 101 andthe relative dielectric constant ε2 of the second cylindrical structuralmember 102. More specifically, a simulation was conducted by changingthe cylindrical radius r1 (=cylindrical radius r2) within a range of 0.2a to 0.3 a and by changing the relative dielectric constant ε2 within arange of 4.0 to 50 for calculating the width of the complete band gap.Results of calculation are summarized in FIG. 5. In the table shown inFIG. 5, a column with “−” means that a complete band gap was not formedfor the cylindrical radius r1 (=cylindrical radius r2) and the relativedielectric constant ε2 corresponding to such column.

As a result of the simulation, in case of a cylindrical radius r1 of0.25 a, the complete band gap showed a width of 12.67% for a relativedielectric constant ε2 of 50. Also in case of a cylindrical radius r1 of0.3 a, the complete band gap showed a width of 3.56% for a relativedielectric constant ε2 of 22, a width of 11.95% for a relativedielectric constant ε2 of 32, and a width of 20.87% for a relativedielectric constant ε2 of 50.

Therefore, a complete band gap could be obtained within a range wherethe radius r1 of the first cylindrical structural member 101 and theradius r2 of the second cylindrical structural member 102 satisfy arelation:0.50 a≦r 1+r 2≦0.60 a,and where the relative dielectric constant ε1 of the first cylindricalstructural member 101 and the relative dielectric constant ε2 of thesecond cylindrical structural member 102 satisfy a relation:ε2/ε1≧20.

Also as in the first example, a complete band gap can be obtained withina frequency range from the millimeter wave to the microwave, by varyingthe lattice constant a of the photonic crystal according to thefrequency (wavelength) of the incident electromagnetic wave.

(Calculation Method for Complete Band Gap)

In the following, there will be explained a method of calculating thewidth of the complete band gap by the aforementioned simulation. Forcalculating the photonic band gap of the photonic crystal embodying thepresent invention, a transmission characteristic simulator “Translight”for the photonic crystal was adopted. This software was developed byAndrew Reynolds during his stay at Glasgow University, and utilizes atransfer matrix method for calculation. It calculates, on a photoniccrystalline structure formed by an assembly of an arbitrary arrangementof cylinders and polygonal prisms, reflection and transmissioncharacteristics for an incident electromagnetic wave (TE wave and TMwave) at an arbitrary incident angle.

Now reference is made to FIG. 1 for explaining the incident angle of theelectromagnetic wave. For the purpose of convenience, a directionperpendicular to a plane on which the cylindrical structural members areperiodically arranged in a tetragonal lattice will be called a z-axisdirection. Also a direction of perpendicular entry of an electromagneticwave 105 into the photonic crystal 100 will be called y-axis, and adirection perpendicular to the y-axis and the z-axis will be called anx-axis. Also an incident angle θ of the electromagnetic wave 105 istaken as 90° when it enters the photonic crystal 100 perpendicularly (iny-axis direction), then becomes smaller towards the x-axis direction andbecomes 0° in a direction parallel to the x-axis. The incident angle ofthe electromagnetic wave 105 can be selected arbitrarily within a rangeof θ=0°-90°, and reflection and transmission characteristics can beobtained for an arbitrary frequency region.

The simulator can calculate the reflection and transmissioncharacteristics of the photonic crystalline structure, by utilizing ashape of the photonic crystalline structure for which the reflection andtransmission characteristics are desired, a frequency region, anincident angle range of the electromagnetic wave (TE wave and TM wave),and relative dielectric constants of the employed materials. In thecalculation, there was employed an incident angle range of 0-90°. As thephotonic crystal 100 embodying the present invention is symmetrical forthe x-y plane, such incident angle range allows calculation for all theincident electromagnetic wave entering from the x-z plane.

A calculation by the simulator provided reflective and transmissionattenuations as a function of the frequency, at each incident angle ofthe electromagnetic wave (TE wave and TM wave). A band gap generationwas recognized in case the transmission attenuation became 20 dB orlarger. In case a band gap is generated at a certain frequency for allthe incident angles (θ=0-90°), a complete band gap is formed at suchfrequency. In case the complete band gap exists continuously overcertain frequencies, a complete band gap width (%) is defined bydividing a range of such frequencies by a central frequency of suchrange. Also in case complete band gaps are present separately indiscrete frequency ranges, a complete band gap width was calculated bysumming the band gap widths present within a normalized frequency rangeof 0.001-1,000.

(Producing Method for Two-Dimensional Photonic Crystal)

In the following, a producing method for the two-dimensional photoniccrystal of the present embodiment will be explained. For example, asimultaneous calcining technology can be utilized in case of producing atwo-dimensional photonic crystal with a ceramic material, and asemiconductor film forming technology can be utilized in case ofproducing a two-dimensional photonic crystal with a semiconductormaterials Also a photoforming method can be utilized in case ofproducing a two-dimensional photonic crystal with a photosettable resin.

At first there will be explained a method for producing atwo-dimensional photonic crystal with a ceramic material. FIG. 6 shows aproducing process of the photonic crystal 100 in the first example. Atfirst, as shown in (a) in FIG. 6, plural green sheets 601 are preparedwith a ceramic material constituting the third dielectric area 103.Then, as shown in (b), the plural green sheets 601 are superposed in ametal mold and laminated by pressing under heating. Then, as shown in(c), the laminated plate is dry etched from above utilizing apredetermined mask, thereby forming plural cylindrical holes 602 andcylindrical holes 603 arranged periodically. In the first example, thecylindrical holes 602 constitute the first dielectric areas, while thecylindrical holes 603 constitute the second dielectric areas.

In the following the second example will be explained with reference toFIG. 7. At first, as shown in (a) in FIG. 7, a green sheet 701 isprepared with a ceramic material for the first cylindrical structuralmember 101 constituting the first dielectric area. Similarly a greensheet 702 is prepared with a ceramic material for the second cylindricalstructural member 102 constituting the second dielectric area. Then, asshown in (b) in FIG. 7, the green sheet 701 for the first cylindricalstructural member and plural green sheets 702 for the second cylindricalstructural member 102 superposed thereon are placed in a metal mold andlaminated by pressing under heating.

Then, as shown in FIG. 7(c) which is a cross-sectional view of (b), thesecond dielectric area is removed by dry etching in a predeterminedarea, utilizing a predetermined mask, until the first dielectric area isexposed, thereby forming plural cylindrical holes 703 arrangedperiodically. Then, as shown in FIG. 7(d), cylinders 704 constitutingthe first dielectric areas are formed by an epitaxial crystal growth inthe cylindrical holes, up to an upper surface. Similarly, cylinders 705constituting the second dielectric areas formed by an epitaxial crystalgrowth in the cylindrical holes. Then, a dry etching is executed with apredetermined mask to form cylinders 704 and cylinders 705 as shown inFIG. 7(e). The cylinders 704 constitute the first cylindrical structuralmembers 101, and the cylinders 705 constitute the second cylindricalstructural members 102.

In the following a producing method for the photonic crystal 100 in thethird example will be explained with reference to FIG. 8. At first, asshown in (a) in FIG. 8, a green sheet 801 is prepared with a ceramicmaterial for the second cylindrical structural member 102 constitutingthe second dielectric area. Similarly a green sheet 802 is prepared witha ceramic material for the third dielectric area 103. Then, as shown in(b) in FIG. 8, the green sheet 801 for the second cylindrical structuralmember and plural green sheets 802 for the third dielectric area 103superposed thereon are placed in a metal mold and laminated by pressingunder heating.

Then, as shown in FIG. 8(c) which is a cross-sectional view of (b), thethird dielectric area is removed by dry etching in a predetermined area,utilizing a predetermined mask, until the material constituting the thesecond dielectric area is exposed, thereby forming plural cylindricalholes 803 arranged periodically. Then, as shown in FIG. 8(d), seconddielectric areas 804 are formed by an epitaxial crystal growth in thecylindrical holes, up to an upper surface. Then, a dry etching isexecuted with a predetermined mask to form plural cylindrical holes 805arranged periodically and constituting the first dielectric areas, asshown in FIG. 8(e).

Thereafter the laminate is divided into a desired shape and calcined toobtain a photonic crystal in which different dielectric members arecalcined simultaneously. Through this process, the first dielectricareas are constituted of air, while the second dielectric areas areformed by the second cylindrical structural members 102 of a ceramicmaterial, and the third dielectric areas 103 are formed therearound witha ceramic material different from that of the second dielectric areas.

In case of producing the two-dimensional photonic crystal of theinvention with a semiconductor material, a mask pattern can be preparedby a photolithographic technology and a desired shape can be obtained bya dry etching.

Also in a photoforming method, a photosettable resin in a liquid stateis irradiated with an ultraviolet beam to cause a polymerizationreaction in the irradiated area only, whereby the photosettable resin ishardened into a desired shape.

Other Embodiments

In the foregoing explanation, the two-dimensional photonic crystal isassumed to have a tetragonal unit lattice, but such case is notrestrictive.

FIG. 9 shows another embodiment of the photonic crystal of theinvention.

A photonic crystal 900 shown in FIG. 9 is constituted of plural firstcylindrical structural members 901, plural second cylindrical structuralmembers 902, and a dielectric area 903 provided around the firstcylindrical structural members 901 and the second cylindrical structuralmembers 902. In this case, the first cylindrical structural member 901constitutes a first dielectric area, and the second cylindricalstructural member 902 constitutes a second dielectric area.

As shown in FIG. 9, the first cylindrical structural members 901 arepositioned at apices of an equilateral triangle and thus constitute atrigonal lattice 904. Thus the photonic crystal 900 has a periodicalstructure in which a trigonal unit lattice 904 is periodically arranged.A second cylindrical structural member 902 is provided at an approximatecenter (vicinity of center of gravity) of each trigonal lattice 904, anda dielectric area 904 is provided around the first cylindricalstructural members 901 and the second cylindrical structural members902. FIG. 9 shows only a part of the photonic crystal 900, but in factthe structure shown in FIG. 9 is periodically arranged.

A complete band gap can be realized even in a trigonal lattice byvarying the dielectric constant ε1 and the radius r1 of the firstcylindrical structural member 901, the dielectric constant ε2 and theradius r2 of the second cylindrical structural member 902, and thedielectric constant ε3 of the third dielectric area 903 under a suitablecondition.

FIG. 10 shows still another embodiment of the photonic crystal of theinvention.

A photonic crystal 950 shown in FIG. 10 is constituted of plural firstcylindrical structural members 951, plural second cylindrical structuralmembers 952, and a dielectric area 953 provided around the firstcylindrical structural members 951 and the second cylindrical structuralmembers 952. In this case, the first cylindrical structural member 951constitutes a first dielectric area, and the second cylindricalstructural member 952 constitutes a second dielectric area.

As shown in FIG. 10, the first cylindrical structural members 951 arepositioned at apices of an equilateral hexagon and thus constitute ahexagonal lattice 954. Thus the photonic crystal 950 has a periodicalstructure in which a hexagonal unit lattice 954 is periodicallyarranged. A second cylindrical structural member 952 is provided at anapproximate center (vicinity of center of gravity) of each hexagonallattice 954, and a dielectric area 953 constituting the third dielectricarea is provided around the first cylindrical structural members 951 andthe second cylindrical structural members 952. FIG. 10 shows only a partof the photonic crystal 950, but in fact the structure shown in FIG. 10is periodically arranged.

A complete band gap can be realized even in a hexagonal lattice byvarying the dielectric constant ε1 and the radius r1 of the firstcylindrical structural member 951, the dielectric constant ε2 and theradius r2 of the second cylindrical structural member 952, and thedielectric constant ε3 of the third dielectric area 903 under a suitablecondition.

As explained in the foregoing, it is rendered possible to form acomplete band gap even in a unit lattice of an equilateral polygonalshape, by suitably regulating the dielectric constant and thecylindrical radius of the first dielectric area, the second dielectricarea and the third dielectric area.

Also the first, second and third dielectric areas are not limited to acylindrical shape but may have a polygonal prism shape. Also the unitlattice is not limited to a shape of an equilateral polygon, but mayhave a shape allowing easy two-dimensional arrangement, and the photoniccrystal may be formed by unit lattices of plural kinds.

APPLICATIONS

The photonic crystal of the present invention is applicable to variousdevices, and the dimension of the device can be made ultra small byutilizing a photonic crystal. For example, the photonic crystal of theinvention can be utilized for producing a waveguide. In such case, byintroducing a linear defect into the photonic crystal, a defect level isformed in the portion of such linear defect, and the electromagneticwave can exist only in such defect level. Therefore a waveguide isformed by a photonic band gap.

In case of an optical waveguide for a light, such optical waveguide hasa dimension of several hundred nanometers or less and can thereforeconfine the light within a space that is smaller by more than 10 timesin comparison with an optical fiber. Also a waveguide prepared with aphotonic crystal does not cause a scattered leakage of light to theexterior even in a steep bend, whereby an ultra small circuit can berealized.

Also the photonic crystal structure of the invention can be applied toan ultra small resonator by introducing a point defect therein, becausethe electromagnetic wave can exist only in a portion of such pointdefect and is confined therein by a surrounding band gap.

INDUSTRIAL APPLICABILITY

The photonic crystal of the invention can realize a complete band gap,by providing a second dielectric area at the approximate center of eachtetragonal lattice which is formed by first dielectric areas and isarranged periodically, and providing a third dielectric areatherearound. Also such photonic crystal can be produced easily as it isa two-dimensional photonic crystal having a tetragonal lattice as a unitlattice.

It is also possible to prepare a two-dimensional photonic crystal havinga wide complete band gap by singly utilizing a material of a relativedielectric constant, particularly a ceramic material.

It is also rendered possible to obtain a complete band gap for theelectromagnetic wave of a wavelength range from a millimeter wave to amicrowave, by varying the lattice constant of the tetragonal latticeaccording to the wavelength of the electromagnetic wave to be used.

It is further rendered possible to obtain a wide complete band gap andto reduce a size of a device utilizing the photonic crystal, byconstituting the periodically arranged tetragonal lattice with adielectric material other than air.

Furthermore, the photonic crystal of the invention can be applied to awaveguide or a resonator, thereby enabling an ultra miniaturization ofthese devices.

1. A two-dimensional photonic crystal formed by a periodicaltwo-dimensional arrangement of plural unit lattices, comprising: aprism-shaped first dielectric area arranged at each lattice point of theunit lattice; a prism-shaped second dielectric area arranged at anapproximate center of the unit lattice; and a third dielectric areaadjacent to and around the first and second dielectric areas.
 2. Atwo-dimensional photonic crystal according to claim 1, characterized inthat the third dielectric area has a relative dielectric constantdifferent from relative dielectric constants of the first and seconddielectric areas.
 3. A two-dimensional photonic crystal according toclaim 2, characterized in that the unit lattice is a tetragonal lattice.4. A two-dimensional photonic crystal according to claim 3,characterized in that the first dielectric area and the seconddielectric area have a substantially cylindrical shape and satisfy arelationship:0.4 a≦r 1+r 2≦0.6 a wherein r1 indicates a radius of the cylindricalfirst dielectric area, r2 indicates a radius of the cylindrical seconddielectric area, and a indicates a unit length of a lattice axis of thetetragonal lattice.
 5. A two-dimensional photonic crystal according toclaim 3, characterized in that a relative dielectric constant ε1 of thefirst dielectric area is equal to a relative dielectric constant ε2 ofthe second dielectric area.
 6. A two-dimensional photonic crystalaccording to claim 3, characterized in that a relative dielectricconstant ε1 of the first dielectric area is smaller than a relativedielectric constant ε2 of the second dielectric area.
 7. Atwo-dimensional photonic crystal according to any one of claims 2 to 5,characterized in that a relative dielectric constant ε3 of the thirddielectric area satisfies at least a relation ε3>ε1.
 8. Atwo-dimensional photonic crystal according to claim 2, characterized inthat a relative dielectric constant ε1 of the first dielectric area, arelative dielectric constant ε2 of the second dielectric area, and arelative dielectric constant ε3 of the third dielectric area satisfyrelations:ε3>ε1, and ε2/ε1>20.
 9. A two-dimensional photonic crystal according toany one of claims 1 to 4, characterized in that the first and seconddielectric areas are formed by air and the third dielectric area isformed by a dielectric material containing a ceramic material.
 10. Atwo-dimensional photonic crystal according to any one of claims 1 to 6,characterized in that the first and second dielectric areas are formedby a dielectric material containing a ceramic material and the thirddielectric area is formed by air.
 11. A two-dimensional photonic crystalaccording to any one of claims 1 to 8, characterized in that the first,second and third dielectric areas are formed by a dielectric materialcontaining a ceramic material.
 12. A two-dimensional photonic crystalaccording to any one of claims 1 to 11, characterized in that a unitlength a of the lattice axis of the tetragonal lattice is differentdepending on a frequency of a light or an electromagnetic wave enteringthe two-dimensional photonic crystal.
 13. A photonic crystal waveguidecharacterized in including a two-dimensional photonic crystal accordingto any of claims 1 to 12, wherein a linear defect is formed in aperiodical lattice arrangement of the two-dimensional photonic crystal.14. A photonic crystal resonator characterized in including atwo-dimensional photonic crystal according to any of claims 1 to 12,wherein a point-shaped defect is formed in a periodical latticearrangement of the two-dimensional photonic crystal.